Mister Exam

Other calculators

(3^n-1)/6^n
Sum of series/ (3^n-1)/6^n

Sum of series (3^n-1)/6^n


∑
=

The solution

You have entered [src] 
  oo        
____        
\   `       
 \     n    
  \   3  - 1
   )  ------
  /      n  
 /      6   
/___,       
n = 0
∑n=0∞3n−16n\sum_{n=0}^{\infty} \frac{3^{n} - 1}{6^{n}} 
Sum((3^n - 1)/6^n, (n, 0, oo))
The radius of convergence of the power series
Given number:
3n−16n\frac{3^{n} - 1}{6^{n}}
It is a series of species
an(cx−x0)dna_{n} \left(c x - x_{0}\right)^{d n}
- power series.
The radius of convergence of a power series can be calculated by the formula:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=3n−1a_{n} = 3^{n} - 1
and
x0=−6x_{0} = -6
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−6+lim⁡n→∞∣3n−13n+1−1∣)\frac{1}{R} = \tilde{\infty} \left(-6 + \lim_{n \to \infty} \left|{\frac{3^{n} - 1}{3^{n + 1} - 1}}\right|\right)
Let's take the limit
we find
False

False

R=0R = 0
The rate of convergence of the power series
0.06.00.51.01.52.02.53.03.54.04.55.05.50.01.0
The answer [src] 
4/5
45\frac{4}{5} 
4/5
Numerical answer [src] 
0.800000000000000000000000000000
0.800000000000000000000000000000
The graph
Sum of series (3^n-1)/6^n

    Examples of finding the sum of a series

    © Mister Exam – Calculator